[10 pts Randomised response is a method used in survey interview that allows respondents to respond...

  

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[10 pts Randomised response is a method used in survey interview that allows respondents to respond to sensitive issues (e.g., are you a smoker?) while maintaining confidentiality. It works as follows. The interviewer prepares two questions: one is the question to be surveyed, while the other is a dummy question whose answer is trivially true (e.g., is 1+1 Each respondent picks a question from the two randomly, and answers 2?). the question truthfully, but the interviewer doesn't know which question is picked. Suppose we have surveyed n people, and know a priori that the number of smokers, denoted m, is no more than n/2. Note that n is known but m is unknown (we only know m < n/2). (a) Suppose a randomly chosen respondent has answered "yes" to the question. Show that the interviewer's confidence on the respondent being a smoker (i.e., the probability the respondent being a smoker given that his or her answer is "yes") is at most 2/3. (b) Suppose X respondents have answered "yes". Design an unbiased estimator of the number of smokers, i.e., find a function f() such that E[f(X)] = m. Justify your answer. [10 pts Randomised response is a method used in survey interview that allows respondents to respond to sensitive issues (e.g., are you a smoker?) while maintaining confidentiality. It works as follows. The interviewer prepares two questions: one is the question to be surveyed, while the other is a dummy question whose answer is trivially true (e.g., is 1+1 Each respondent picks a question from the two randomly, and answers 2?). the question truthfully, but the interviewer doesn't know which question is picked. Suppose we have surveyed n people, and know a priori that the number of smokers, denoted m, is no more than n/2. Note that n is known but m is unknown (we only know m < n/2). (a) Suppose a randomly chosen respondent has answered "yes" to the question. Show that the interviewer's confidence on the respondent being a smoker (i.e., the probability the respondent being a smoker given that his or her answer is "yes") is at most 2/3. (b) Suppose X respondents have answered "yes". Design an unbiased estimator of the number of smokers, i.e., find a function f() such that E[f(X)] = m. Justify your answer.